Solution

The sum of the circumferences of two circles is \(\quantity{60\pi}{cm}\). and the sum of their areas is \(\quantity{458\pi}{sq.cm.}\) Calculate the radii of the circles.

If a circle has radius \(\quantity{x}{cm}\), then its circumference is \(\quantity{2\pi x}{cm}\) and its area is \(\quantity{\pi x^2}{sq.cm.}\)

Let us denote the radii of the two circles in the question by \(\quantity{r}{cm}\) and \(\quantity{R}{cm}\). The information in the question leads to two equations:
\[\begin{align*}
2\pi r + 2\pi R &= 60 \pi, \\
\pi r^2 + \pi R^2 &= 458 \pi.
\end{align*}\]
By dividing the first equation through by \(2\pi\) and the second by \(\pi\), these equations become
\[\begin{align*}
r + R &= 30, \\
r^2 + R^2 &= 458.
\end{align*}\]

If \(r = R\), then \(r = R = 15\), and the second equation does not hold. So \(r \neq R\), and we can say \(R > r\).